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# 100wk12solnsf13.pdf

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Wilfrid Laurier University

Mathematics

MA100

Christine Zaza

Fall

Description

MA100 Week 12 Report - Graphing
Name: Student Number: Lab Fall 2013
1. [4 marks] Suppose f(x) = x + 5x . Determine the absolute maximum and the absolute minimum of f for
x 2 [▯2;2].
f (x) = 0 () 5x + 20x = 0 Now: f(▯2) = ▯32 + 5(16) = 48
() 5x (x + 4) = 0 f(0) = 0 ! absolute minimum
() x = 0 since x = ▯4 2= [▯2;2] f(2) = 32 + 5(16) = 112 ! absolute maximum
ex
2. [22 marks] Consider the function f(x) = :
x + 1
(a) State the domain of f using interval notation. dom f (▯1;▯1) [ (▯1;1)
(b) Determine any x-intercepts and y-intercepts of f.
y-int ) f(0) = 1. There is a y-int at (0;1). There are no x-int as f(x) 6= 0.
(c) Consider lim f(x) and lim f(x) and comment on the vertical asymptotes of f.
x!▯1 + x!▯1▯
ex ex
lim▯ f(x) = lim ▯ lim+ f(x) = lim+
x!▯1 x!▯1 x + 1 x!▯1 x!▯1 x + 1
▯1 ▯1
e e
+ = +1 ▯ = ▯1
0 0
) There is a vertical asymptote at x = ▯1
e ex
(d) Given that lim = 1 and lim = 0, ▯nd all horizontal asymptotes of f (if any). Justify your
x!1 x x!▯1 x
answer.
x x
lim f(x) = lim e lim f(x) = lim e
x!1 x!1 x + 1 x!▯1 x!▯1 x + 1
ex ▯ ▯ ex ▯ ▯
= lim x approaches 1 = lim x approaches 0
x!1 1 + 1 1 x!▯1 1 + 1 1
x x
= 1 = 0
) y = 0 is the only H.A.
x x▯ 2 ▯
(e) Show that f (x) = xe an

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